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On-brane data for braneworld stars
Matt Visser∗
School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand
David L. Wiltshire†
arXiv:hep-th/0212333v2 20 Feb 2003
Department of Physics & Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
(Dated: 30 December 2002; Revised 20 February 2003; LATEX-ed February 1, 2008)
Stellar structure in braneworlds is markedly different from that in ordinary general relativity. As
an indispensable first step towards a more general analysis, we completely solve the “on brane”
4-dimensional Gauss and Codazzi equations for an arbitrary static spherically symmetric star in a
Randall–Sundrum type II braneworld. We then indicate how this on-brane boundary data should be
propagated into the bulk in order to determine the full 5-dimensional spacetime geometry. Finally,
we demonstrate how this procedure can be generalized to solid objects such as planets.
PACS numbers: 04.70.Dy, 04.62.+v,11.10.Kk
Keywords: Braneworld, stars, black holes, hep-th/0212333
I.
INTRODUCTION
One physically important problem in the braneworld
scenario is the development of a full understanding of
stellar structure and black holes [1, 2, 3, 4, 5]. What
is already known is that stellar structure in braneworlds
is rather different from that in ordinary general relativity. The key point is that if one is confined to making
physical measurements on the brane, then the restricted
“on-brane” version of the Einstein equations does not
form a complete system for specifying the brane geometry [6, 7, 8]. The problem lies in the fact that the bulk 5dimensional Weyl tensor feeds into the “on-brane” equations and so connects the brane to the bulk.
On the other hand, one may solve the “on-brane”
version of the Einstein equations to obtain a consistent
class of boundary data that satisfies the 4-dimensional
Gauss and Codazzi equations [6], analogously to the initial data in the standard (3 + 1) decomposition of globally hyperbolic manifolds in general relativity. Although
the relevant partial differential equations are now elliptic
rather than hyperbolic, one can use the boundary data
in a completely analogous fashion — as input into the
5-dimensional bulk Einstein equations in order to “propagate” the 4-dimensional geometry off the brane and into
the bulk [1, 5].
In this Letter we present an algorithm for completely
solving the 4-dimensional Gauss and Codazzi equations
for a static spherically symmetric star on the brane. This
provides the most general boundary data, suitable for
then determining the bulk geometry by an appropriate
relaxation method [1].
Specifically, we will use Gaussian normal coordinates
adapted to a timelike 4-dimensional hypersurface (spacelike normal) in a 5-dimensional geometry,
ds5 2 = dη 2 + gab dxa dxb
(1)
and assume a type II Randall–Sundrum braneworld [9],
in which the bulk metric is a 5-dimensional Einstein
space, (i.e., with Ricci tensor proportional to the metric), but the 4-dimensional Lorentzian signature metric
gab is as yet undetermined.
If we impose ZZ2 symmetry on this spacetime, and tune
the 5-dimensional bulk cosmological constant and brane
tension appropriately, then the junction conditions together with the projected 5-dimensional Einstein equations (the Gauss equations) reduce to the induced but incomplete on-brane “Einstein equations” in the form [2, 6]
Gab = 8π Tab − Λ4 gab − Eab
(2)
2 κ
(T 2 )ab − 13 T Tab − 21 gab T · T − 31 T 2 .
−
4
where κ2 is a constant inversely proportional to the brane
tension, Λ4 is an on-brane cosmological term, the nonlocal term Eab is simply the projection of the 5-dimensional
Weyl tensor onto the brane, and in addition to the
usual stress-energy term there is also a nonlinear term
quadratic in stress-energy. In addition to (2) one finds
that the extrinsic curvature
Kab =
1 ∂gab
,
2 ∂η
(3)
is related to the on-brane fields via
Kab = −
8π
κ Tab − 13 gab T −
gab .
2
κ
(4)
The Codazzi equation
∗ Electronic
address:
[email protected];
URL: http://www.mcs.vuw.ac.nz/~visser
† Electronic
address:
[email protected] ;
URL: http://www.phys.canterbury.ac.nz/~physdlw/
∇b [K ab − Kg ab ] = 0,
(5)
is then equivalent to 4-dimensional stress-energy conservation for the on-brane matter: ∇a Tab = 0. Eqs. (2)
2
and (4) complete the specification of the boundary data
by effectively supplying the “on brane” metric and its
normal derivative.
The only truly general thing we know about the nonlocal Weyl tensor projection term Eab is that it is traceless,
E a a = 0. Together with (2) this implies that
R = −8π T + 4Λ4 − 41 κ2 T · T − 13 T 2 .
(6)
In the vacuum region outside a star this reduces to
R = 4Λ4 ,
(7)
whereas inside the star it takes the form
R = [nonlinear-source].
(8)
If for instance we are dealing with a perfect fluid then
R = 8π(ρ − 3p) + 4Λ4 + 14 κ2 (ρ2 − 3p2 ) + 31 (ρ − 3p)2 .
(9)
If we are restricted to making physical measurements
on the brane, then (6) is the only really general thing
we can say. Eqs. (2), (6) are much weaker than the 4dimensional Einstein equations and so the solution space
will be much more general. Of course, we must also
use the extrinsic curvature constraint (4) to complete
the boundary data, and then ultimately probe the bulk
geometry off the brane, and this will indirectly provide
further restrictions.
Our aim here is to solve Eqs. (2) and (4) in full generality for arbitrary static spherically symmetric solutions
on the brane, to provide appropriate data to “propagate”
into the bulk [1] (via Eqs. (32) below), to thereby test the
consistency of candidates for realistic stellar models.
II.
VACUUM [Λ4 = 0]
In this section we will consider the case when Λ4 = 0.
For a braneworld star, in the vacuum region outside the
surface we are interested in solving R = 0. In the usual
way we can locally adopt on-brane coordinates such that
ds2 = − exp[−2φ(r)] dt2 +
dr2
+ r2 dΩ2 .
B(r)
(10)
If we now impose the condition R = 0 we have one differential constraint connecting two unknowns — therefore
there will be a nondenumerable infinity of solutions parameterized by some arbitrary function of r. Various
specific cases have already been discussed in the literature, but no attempt has been made at extracting the
general solution.
A.
Special cases
Known special case solutions include:
— “Reissner-Nordström-like” [3]
B(r) = exp[−2φ(r)] = 1 −
Z
2M
+ 2.
r
r
(11)
Note that the parameter Z is not an electric charge, but
should be thought of as a tidal distortion parameter.
— “Spatial Schwarzschild” [4]
B(r) = 1 −

ǫ +
exp[−2φ(r)] = 

2M (1 + ǫ)
;
r
r
2
2M (1 + ǫ)

r
 .

1+ǫ
(12)
1−
(13)
(This geometry is also discussed in a rather different context in ref. [10].)
— “Temporal Schwarzschild” [3, 4]
!
2M
a
B(r) = 1 −
1+
;
(14)
r
r(1 − 23 M
r )
exp[−2φ(r)] = 1 −
2M
.
r
(15)
However these are all very specific special cases; and are
in no way general.
B.
General vacuum solution
Let us first write the metric in the form
Z ∞
dr2
2
+ r2 dΩ2 .
ds = − exp −2
g(r̄) dr̄ dt2 +
B(r)
r
(16)
The function g(r) is interpreted as the locally-measured
acceleration due to gravity; it is positive for a inward
acceleration.
Now calculate the Ricci scalar R
(2r + r2 g)B ′ + (2r2 g 2 + 2r2 g ′ + 4rg + 2)B − 2
.
r2
(17)
For the vacuum case we set this equal to zero, which
yields
R=
(2r + r2 g)B ′ + (2r2 g 2 + 2r2 g ′ + 4rg + 2)B − 2 = 0. (18)
If we treat g(r) as input and view this as a differential
equation for B(r), it is a first-order linear ODE, and
hence explicitly solvable. The integrating factor is
Z r
1 + 2r̄g(r̄) + r̄2 g(r̄)2 + r̄2 g ′ (r̄)
F (r; r0 ) = exp
dr̄ ,
r̄(1 + r̄g(r̄)/2)
r0
(19)
where r0 is any convenient reference point, and the general solution is
Z r
F (r̄; r0 )
dr̄ + B(r0 ) F (r; r0 )−1 .
B(r) =
r0 r̄ (1 + r̄g(r̄)/2)
(20)
3
Whereas r0 is an arbitrary gauge parameter, the constant B(r0 ) is related to physical parameters such as the
mass and post-Newtonian corrections. In the case of the
“temporal Schwarzschild” solution (14), (15), for example, B(r0 ) = 2(a + r0 ) − 3M .
Alternatively one may use B(r) as input to generate a
first-order ODE quadratic in g(r) — a Riccati equation.
However, this has no comparable general solution. The
algorithm above is a modification of Lake’s construction
for generating spherically symmetric perfect fluid spacetimes [11]. We suspect that an isotropic coordinate version, based on [12], may also be viable.
This general solution can be somewhat simplified by
using integration by parts on the integrating factor to
remove the derivatives of g(r). We find
F (r; r0 ) =
(1 + rg(r)/2)
2
(21)
2
(1 + r0 + g(r0 )/2)
Z r
1 + r̄g(r̄) + r̄2 g(r̄)2
dr̄ ,
× exp
r̄(1 + r̄g(r̄)/2)
r0
2
r 1 + 21 rg(r)
=
(22)
2 F2 (r; r0 ),
r0 1 + 21 r0 g(r0 )
where we have defined
Z
F2 (r; r0 ) ≡ exp
r
r0
There is a potential subtlety if the Schwarzschild coordinate r does not increase monotonically with respect
to the outward radial proper distance, ℓr . For ordinary
stars in general relativity the monotonicity of r with respect to ℓr is guaranteed by the null energy condition. In
braneworld stars there is no particular reason to believe
in the monotonicity of r(ℓr ), but our construction will
still hold piecewise on monotonic intervals.
III.
We now want to solve R = S(r), with S(r) a specified source. We can now consider arbitrary values of Λ4
without additional complications by including a possibly
non-zero Λ4 in S(r). Proceeding exactly as above we find
Z r
1 + r̄2 S(r̄)/2
B(r) =
F (r̄; r0 ) dr̄ + B(r0 )
r0 r̄ [1 + r̄g(r̄)/2]
×F (r; r0 )−1 ,
with the same integrating factor F (r; r0 ) of Eq. (21).
Equivalently
(Z
r
B(r) =
1 + 21 r̄2 S(r̄) 1 + 12 r̄g(r̄) F2 (r̄) dr̄
(23)
+r0 1 +
2
1
2 r0 g(r0 )
)
B(r0 )
o−1
n 2
.
× r 1 + 12 rg(r) F (r; r0 )
r
(1 + r̄g(r̄)/2) F2 (r̄; r0 ) dr̄
r0
)
2
+r0 1 + 21 r0 g(r0 ) B(r0 )
o−1
n 2
.
× r 1 + 21 rg(r) F2 (r; r0 )
(24)
This is now the explicit general (static spherically symmetric) solution of the equation R = 0 using the function
g(r) and the constant B(r0 ) as arbitrary input. The construction is completely algorithmic.
T ab
(25)
r0
g(r̄)[1 + 2r̄g(r̄)]
dr̄ .
2(1 + 21 r̄g(r̄))
Eq. (20) then becomes
(Z
B(r) =
STELLAR INTERIOR
(26)
Once the source S(r) is specified, this is fully general. In
addition one must specify an arbitrary function g(r), and
a single arbitrary constant B(r0 ), and so algorithmically
determine the metric on the brane.
Of course the source S(r) is somewhat restricted in
that it is an algebraic function of the on-brane stressenergy tensor, which is itself restricted by 4-dimensional
energy-momentum conservation (equivalent to the Codazzi equation (5) as mentioned above). For a static
perfect fluid with spherical symmetry the stress-energy
has the form

R∞
ρ(r) exp +2 r g(r̄) dr̄
0
0
0


0
p(r) B(r)
0
0
.
=
2


0
0
p(r)/r
0
2
2
0
0
0
p(r)/(r sin θ)

(27)
The equation of energy–momentum conservation gives
dp
= −g(r) [ρ + p],
dr
(28)
4
which, being a linear first-order ODE, has the exact closed-form solution
Z
p(r) = exp −
r
r0
Z
g(r̄) dr̄ × p(r0 ) −
The lesson now is that to find all possible conserved
tensors T ab one is free to specify the function ρ(r) and
the number p(r0 ) arbitrarily, and thereby calculate p(r),
which now yields the full tensor T ab . From T ab (r) we
now calculate S(r), thereby fixing the intrinsic geometry
on the brane. By eq. (4) this also automatically generates
the most general possible candidate for the extrinsic curvature K ab compatible with the assumed symmetries. It
must be emphasised that only some of these braneworld
geometries are physically meaningful, because one now
needs to extrapolate them off the brane to see if the
“graviton” is still bound [13].
IV.
EXTRAPOLATING OFF THE BRANE
The full algorithm is:
• Step 1: Pick an arbitrary density distribution ρ(r)
of matter inside the star; an arbitrary function g(r);
and a single number p(r0 ). Calculate p(r) and so
evaluate the stress-energy tensor T ab and the source
term
S(r) = −8πT + 4Λ4 − 41 κ2 T · T − 31 T 2
(30)
• Step 2: Armed with this source term S(r) and the
previously chosen g(r), pick one additional number
B(r0 ) in order to calculate the function B(r) and
thereby generate a candidate metric gab for the onbrane physics.
By calculating the 4-dimensional Einstein tensor
for this candidate metric, one can rearrange Eq. (2)
to calculate Eab = (5) Cηaηb and so the find the projection of the five-dimensional Weyl tensor on the
brane.
• Step 3: Using this on-brane metric, and the onbrane stress-energy calculated in Step 1, use Eq (4)
to calculate the extrinsic curvature Kab .
• Step 4: Evolve the metric off the brane. Sufficiently
near the brane one can certainly use normal coor-
T ab
r
r0
Z
g(r̄) ρ(r̄) exp +
r̄
g(r̃) dr̃ dr̄ .
r0
(29)
dinates and so the standard result
(5)
Rηaηb =
∂Kab
+ Kam K m b
∂η
(31)
applies. Rearrange this to yield:
1 ′ ′ ∂gaa′ ∂gb′ b
∂ 2 gab
= − ga b
+ 2 (5) Rηaηb .
∂η 2
2
∂η
∂η
(32)
Here (5) Rηaηb is calculable in terms of the projection of the five-dimensional Weyl tensor and the
bulk cosmological constant.
This now provides a well-determined set of equations for
extending the on-brane metric into the bulk. Whether
or not the resulting braneworld model is actually viable
depends on whether or not the bulk geometry is “well
behaved” — in particular, is the graviton bound or unbound [13].
We must make the caveat that the Gaussian normal
coordinates system (1) is likely to break down at some
stage as one moves away from the brane. This is not
really a fundamental objection but more of a technical
issue. While it is easiest to set up the “on-brane” boundary conditions using Gaussian normal coordinates, as a
practical matter when it comes to numerically solving for
the bulk geometry one should be prepared to dynamically
adjust the coordinate system.
V.
SOLID PLANETS
If we are dealing with a situation of spherical symmetry
that does not correspond to a perfect fluid, such as a
solid planet, the radial and transverse pressures would
be different and we would have
R = 8π(ρ − pr − 2pt ) + 4Λ4
(33)
2
2
2
2
1 2
1
+ 4 κ (ρ − pr − 2pt ) + 3 (ρ − pr − 2pt ) .
The other major change arises in the on-brane stressenergy tensor, which becomes
R∞
ρ(r) exp +2 r g(r̄) dr̄
0
0
0

0
p
(r)
B(r)
0
0
r
=

0
0
pt (r)/r2
0
0
0
0
pt (r)/(r2 sin2 θ)



.

(34)
5
Then covariant conservation gives the slightly more complicated equation
2
dpr
= (pt − pr ) − g(r) [ρ + pr ].
dr
r
This is still a linear first-order ODE, and has the exact closed-form solution
Z r̄
Z r
Z r
2
2
−2
g(r̃) dr̃ 2 r̄pt (r̄) − g(r̄) ρ(r̄) r̄ dr̄ + r0 pr (r0 ) .
exp +
g(r̄) dr̄ ×
pr (r) = r exp −
r0
r0
The lesson now is that to find all possible conserved tensors T ab one is free to specify the functions ρ(r), pt (r),
g(r), and the number pr (r0 ) arbitrarily, and thereby calculate pr (r) which now yields the full stress-energy tensor T ab . This is now used to calculate S(r). Once B(r0 )
is specified the on-brane intrinsic geometry (encoded in
B(r)) is calculable. Ipso facto, this procedure also generates the most general possible candidate for the extrinsic
curvature K ab compatible with the assumed symmetries.
Thus spherically symmetric solid objects such as planets
are not much more difficult to deal with than are fluid
objects such as stars.
VI.
CONCLUSION
What we have done in this article is to find the most
general set of “on-brane data” suitable for characterizing an arbitrary static spherically symmetric braneworld
star. This on-brane data is characterized by two arbitrary functions g(r), and ρ(r), and two arbitrary constants B(r0 ), and p(r0 ). This is an enormous dataset,
much more general than the various special cases discussed in [3, 4, 5]. After suitable notational modifications, this boundary data can be used as input into Wiseman’s relaxation algorithm for determining the bulk geometry [1]. Although Wiseman’s algorithm [1] is general,
his numerical examples were restricted to special choices
of boundary data. Hopefully, by characterizing the most
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arXiv:gr-qc/0111065.
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(35)
(36)
r0
general set of boundary data, we are providing the ingredients for a more systematic treatment of the numerical
problem in future work.
Ultimately, it will be the bulk geometry that determines whether or not a specific set of on-brane data leads
to a physically meaningful star [13]. We believe the technique we have developed here could be quite powerful
because it is completely algorithmic. Ideally, we would
hope that broad statements about the class of possible
stellar structure functions S(r) could made. For example, by a suitable characterization of possible pathological
singularities in the bulk geometry, it might be possible
to rule out particular classes of structure functions.
We conclude that braneworld stars are potentially
much more complicated than standard general relativistic stars, and emphasise that the coupling to the bulk will
play an important role in restricting the possible functions g(r) which enter the Tolman-Oppenheimer-Volkofflike equation (28).
Acknowledgement: This work was supported by the
Marsden Fund of the Royal Society of New Zealand.
Note added: After this work was completed a paper by
Bronnikov and Kim appeared [14] which also derives the
pure vacuum solution (21)–(24), but in the context of
braneworld wormholes. That analysis does not consider
the presence of matter, nor does it consider the extrinsic
curvature.
(2000) 023 [arXiv:hep-th/0002091].
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Rev. D 65 (2002) 064004 [arXiv:gr-qc/0109069].
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solutions of Einstein’s Equations”, arXiv:gr-qc/0209104.
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(2002) 935 [arXiv:gr-qc/0103065].
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a brane world”, arXiv:gr-qc/0212112.